Edge-Diameter of a Graph and Its Longest Cycles

Given a graph G and X, Y⊂ V(G) , d_G(X, Y) is the distance between X and Y and the edge diameter diam_e(G) is the greatest distance between two edges of G. In this note, we consider edge diameter of a graph and its longest cycles and prove the following: (1)Let G be a connected graph other than a tree with diam_e(G) ≤ d^′ , then G has a longest cycle D such that d_G(e, D) ≤ d^′- 1 for any edge e of G, furthermore, if G is 2-connected, then d_G(e, C) ≤ d^′- 1 for any longest cycle C and any edge e of G.(2)Let H be a 3-connected simple graph with diam_e(H) ≥ d^′ . Then H has a cycle of length at least 2 d^′+ 3 if H is not K₄ , furthermore, H has a cycle of length at least 2 d^′+ 4 if d^′≥ 4 .